\(\Lambda_{24}\) Leech lattice-shell code

\(\Lambda_{24}\) Leech lattice-shell code[1]

Description

Spherical code whose codewords are points on the \(\Lambda_{24}\) Leech lattice normalized to lie on the unit sphere. The minimal shell of the lattice yields the \((24,196560,1)\) code, and recursively taking its kissing configurations yields the \((23,4600,1/3)\) and \((22,891,1/4)\) spherical codes [2]. The \((24,196560,1)\) and \((23,4600,1/3)\) codes are optimal and unique for their parameters [3]. The \((22,891,1/4)\) code is optimal and unique for its parameters [4,5]. Further recursive kissing configurations yield the \((21,336,1/5)\) and \((20,170,1/6)\) spherical codes [3].

Protection

The minimal-shell code yields an optimal solution to the kissing problem in 24D [3]. This code saturates the Levenshtein bound [6–11][12; pg. 337] and is unique up to equivalence [3]. Its kissing configuration also yields a unique \((23,4600,1/3)\) spherical code [3].

Cousins

\(\Lambda_{24}\) Leech lattice— The \(\Lambda_{24}\) lattice-shell code is obtained from a shell of the Leech lattice.
Spherical sharp configuration— The smallest-shell \((24,196560,1)\) code is a spherical sharp configuration [13,14]. The \((23,4600,1/3)\) [3–5,14][15; Table 1] and \((22,891,1/4)\) [4,5,14][15; Table 1] spherical codes are also sharp configurations.
Spherical design— The smallest-shell \((24,196560,1)\) code is a tight and unique spherical 11-design [3][12; Ch. 3]. The \((23,4600,1/3)\) [3–5,14][15; Table 1] and \((22,891,1/4)\) [4,5,14][15; Table 1] spherical codes are also sharp configurations.
McLaughlin spherical code— The \((23,552,1/5)\) McLaughlin code can be derived from a shell of the Leech lattice [14,16].

Member of code lists

Classical codes

Primary Hierarchy

Classical Domain

Spherical Kingdom

Constant-energy spherical codeECC

Spherical code

Lattice-shell code

Parents

Lattice-shell code

\(\Lambda_{24}\) Leech lattice-shell code

References

[1]: J. Leech, “Notes on Sphere Packings”, Canadian Journal of Mathematics 19, 251 (1967) DOI
[2]: P. Delsarte, J. M. Goethals, and J. J. Seidel, “Spherical codes and designs”, Geometriae Dedicata 6, 363 (1977) DOI
[3]: E. Bannai and N. J. A. Sloane, “Uniqueness of Certain Spherical Codes”, Canadian Journal of Mathematics 33, 437 (1981) DOI
[4]: R. A. Wilson, “Vector stabilizers and subgroups of Leech lattice groups”, Journal of Algebra 127, 387 (1989) DOI
[5]: H. Cohn and A. Kumar, “Uniqueness of the (22,891,1/4) spherical code”, (2007) arXiv:math/0607448
[6]: V. I. Levenshtein, “On choosing polynomials to obtain bounds in packing problems.” Proc. Seventh All-Union Conf. on Coding Theory and Information Transmission, Part II, Moscow, Vilnius. 1978
[7]: V. I. Levenshtein, “On bounds for packings in n-dimensional Euclidean space”, Doklady Akademii Nauk SSSR, 245:6 (1979), 1299–1303
[8]: V. I. Levenshtein. “Bounds for packings of metric spaces and some of their applications”. Problemy Kibernetiki, 40 (1983), 43-110
[9]: V. I. Levenshtein, “Designs as maximum codes in polynomial metric spaces”, Acta Applicandae Mathematicae 29, 1 (1992) DOI
[10]: V. I. Levenshtein, “Universal bounds for codes and designs,” in Handbook of Coding Theory, Vol. I, Part 1, eds. V. S. Pless and W. C. Huffman (Elsevier, 1998), pp. 499-648
[11]: A. M. Odlyzko and N. J. A. Sloane, “New bounds on the number of unit spheres that can touch a unit sphere in n dimensions”, Journal of Combinatorial Theory, Series A 26, 210 (1979) DOI
[12]: J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
[13]: N. N. Andreev, “Location of points on a sphere with minimal energy”. (Russian) Tr. Mat. Inst. Steklova 219 (1997), Teor. Priblizh. Garmon. Anal., 27–31; translation in Proceedings of the Steklov Institute of Mathematics 1997, no. 4(219), 20–24
[14]: H. Cohn and A. Kumar, “Universally optimal distribution of points on spheres”, Journal of the American Mathematical Society 20, 99 (2006) arXiv:math/0607446 DOI
[15]: H. Cohn, “Packing, coding, and ground states”, (2016) arXiv:1603.05202
[16]: M. Dutour Sikirić, A. Schürmann, and F. Vallentin, “The Contact Polytope of the Leech Lattice”, Discrete & Computational Geometry 44, 904 (2010) arXiv:0906.1427 DOI

Page edit log

Victor V. Albert (2022-11-17) — most recent

Cite as:

“\(\Lambda_{24}\) Leech lattice-shell code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/leech_shell

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/spherical/lattice_shell/leech_shell.yml.